determining the strength of a regression model\njanelle tracks the number of miles she drives and the number…

Explanation:

Step1: Label variables

Let $x$ be miles - driven and $y$ be gallons - in - tank. Calculate the necessary sums: Let $n = 6$. $\sum_{i = 1}^{n}x_{i}=27 + 65+83 + 109+142+175=591$ $\sum_{i = 1}^{n}y_{i}=13 + 12+11 + 10+9+8=63$ $\sum_{i = 1}^{n}x_{i}^{2}=27^{2}+65^{2}+83^{2}+109^{2}+142^{2}+175^{2}=27^{2}+4225+6889+11881+20164+30625=73863$ $\sum_{i = 1}^{n}y_{i}^{2}=13^{2}+12^{2}+11^{2}+10^{2}+9^{2}+8^{2}=169+144+121+100+81+64=679$ $\sum_{i = 1}^{n}x_{i}y_{i}=27\times13+65\times12+83\times11+109\times10+142\times9+175\times8=351+780+913+1090+1278+1400=5812$

Step2: Calculate the slope $b_1$

The formula for the slope $b_1$ of the regression line $y = b_0 + b_1x$ is $b_1=\frac{n\sum_{i = 1}^{n}x_{i}y_{i}-\sum_{i = 1}^{n}x_{i}\sum_{i = 1}^{n}y_{i}}{n\sum_{i = 1}^{n}x_{i}^{2}-(\sum_{i = 1}^{n}x_{i})^{2}}$ $n\sum_{i = 1}^{n}x_{i}y_{i}-\sum_{i = 1}^{n}x_{i}\sum_{i = 1}^{n}y_{i}=6\times5812-591\times63=34872 - 37233=-2361$ $n\sum_{i = 1}^{n}x_{i}^{2}-(\sum_{i = 1}^{n}x_{i})^{2}=6\times73863-591^{2}=443178-349281 = 93897$ $b_1=\frac{-2361}{93897}\approx - 0.0252$

Step3: Calculate the intercept $b_0$

The formula for the intercept $b_0$ is $b_0=\bar{y}-b_1\bar{x}$, where $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}=\frac{591}{6}=98.5$ and $\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}=\frac{63}{6}=10.5$ $b_0 = 10.5-(-0.0252)\times98.5=10.5 + 2.4822=12.9822$ The linear - regression model is $y=-0.0252x + 12.9822$

Step4: Calculate the correlation coefficient $r$

The formula for the correlation coefficient $r$ is $r=\frac{n\sum_{i = 1}^{n}x_{i}y_{i}-\sum_{i = 1}^{n}x_{i}\sum_{i = 1}^{n}y_{i}}{\sqrt{(n\sum_{i = 1}^{n}x_{i}^{2}-(\sum_{i = 1}^{n}x_{i})^{2})(n\sum_{i = 1}^{n}y_{i}^{2}-(\sum_{i = 1}^{n}y_{i})^{2})}}$ $n\sum_{i = 1}^{n}y_{i}^{2}-(\sum_{i = 1}^{n}y_{i})^{2}=6\times679 - 63^{2}=4074-3969 = 105$ $r=\frac{-2361}{\sqrt{93897\times105}}=\frac{-2361}{\sqrt{9859185}}\approx\frac{-2361}{3139.93}\approx - 0.752$

Step5: Determine the strength of the model

The absolute value of the correlation coefficient $|r| = 0.752$. Since $0.7\leq|r|\lt0.9$, the strength of the model is strong.

Answer:

Linear regression model: $y=-0.0252x + 12.9822$ Correlation coefficient: $-0.752$ Strength of the model: Strong